The diagram you have shared contains a composite figure of two triangles, with angles and side lengths labeled. The task is to find the measures of certain angles ($\angle 2$, $\angle 4$, $\angle 5$, $\angle 6$, and $\angle 7$) based on the given information.

Here’s how we can approach it:

1. Use known angles: The figure gives us the following information about the angles:

   • $\angle 1 = 30^\circ$
   • $\angle 3 = 70^\circ$
   • $\angle 8 = 10^\circ$

2. Identify triangle properties: Since we are dealing with triangles, the sum of interior angles in any triangle is $180^\circ$. Additionally, one of the triangles includes a right angle ($90^\circ$).

3. Step-by-step solution:

   • For $\angle 2$: This angle is supplementary to $\angle 1$ because they form a straight line. Thus, $\angle 2 = 180^\circ - \angle 1 = 180^\circ - 30^\circ = 150^\circ.$
   • For $\angle 4$: $\angle 4$ is part of a triangle with $\angle 7 = 90^\circ$ and the given $\angle 8 = 10^\circ$. Using the triangle sum property: $\angle 4 = 180^\circ - (90^\circ + 10^\circ) = 80^\circ.$
   • For $\angle 5$: Since $\angle 3 = 70^\circ$ and $\angle 5$ are supplementary (on a straight line), we have: $\angle 5 = 180^\circ - 70^\circ = 110^\circ.$
   • For $\angle 6$: This is an exterior angle of the triangle involving $\angle 5 = 110^\circ$, so: $\angle 6 = \angle 5 = 110^\circ.$
   • For $\angle 7$: This is a right angle, so: $\angle 7 = 90^\circ.$

Final Angle Measures:

• $\angle 2 = 150^\circ$
• $\angle 4 = 80^\circ$
• $\angle 5 = 110^\circ$
• $\angle 6 = 110^\circ$
• $\angle 7 = 90^\circ$

Would you like more details on any of these calculations?

Here are 5 related questions:

1. How can we determine unknown angles in more complex polygons?
2. What is the sum of interior angles for polygons with more than 3 sides?
3. How do supplementary angles help solve for missing values in geometry problems?
4. What properties do exterior angles of triangles possess?
5. How do right angles affect the calculations in composite geometric figures?

Tip: Always remember that the sum of angles in a triangle is 180°, and this can help solve many geometry problems efficiently.